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| Mirrors > Home > MPE Home > Th. List > ellspsn6 | Structured version Visualization version GIF version | ||
| Description: Relationship between a vector and the 1-dim (or 0-dim) subspace it generates. (Contributed by NM, 8-Aug-2014.) (Revised by Mario Carneiro, 8-Jan-2015.) | 
| Ref | Expression | 
|---|---|
| ellspsn5b.v | ⊢ 𝑉 = (Base‘𝑊) | 
| ellspsn5b.s | ⊢ 𝑆 = (LSubSp‘𝑊) | 
| ellspsn5b.n | ⊢ 𝑁 = (LSpan‘𝑊) | 
| ellspsn5b.w | ⊢ (𝜑 → 𝑊 ∈ LMod) | 
| ellspsn5b.a | ⊢ (𝜑 → 𝑈 ∈ 𝑆) | 
| Ref | Expression | 
|---|---|
| ellspsn6 | ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ellspsn5b.a | . . . 4 ⊢ (𝜑 → 𝑈 ∈ 𝑆) | |
| 2 | ellspsn5b.v | . . . . 5 ⊢ 𝑉 = (Base‘𝑊) | |
| 3 | ellspsn5b.s | . . . . 5 ⊢ 𝑆 = (LSubSp‘𝑊) | |
| 4 | 2, 3 | lssel 20936 | . . . 4 ⊢ ((𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) | 
| 5 | 1, 4 | sylan 580 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑉) | 
| 6 | ellspsn5b.w | . . . . 5 ⊢ (𝜑 → 𝑊 ∈ LMod) | |
| 7 | 6 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑊 ∈ LMod) | 
| 8 | 1 | adantr 480 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑈 ∈ 𝑆) | 
| 9 | simpr 484 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → 𝑋 ∈ 𝑈) | |
| 10 | ellspsn5b.n | . . . . 5 ⊢ 𝑁 = (LSpan‘𝑊) | |
| 11 | 3, 10 | lspsnss 20989 | . . . 4 ⊢ ((𝑊 ∈ LMod ∧ 𝑈 ∈ 𝑆 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) | 
| 12 | 7, 8, 9, 11 | syl3anc 1372 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑁‘{𝑋}) ⊆ 𝑈) | 
| 13 | 5, 12 | jca 511 | . 2 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑈) → (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) | 
| 14 | 2, 10 | lspsnid 20992 | . . . . 5 ⊢ ((𝑊 ∈ LMod ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) | 
| 15 | 6, 14 | sylan 580 | . . . 4 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → 𝑋 ∈ (𝑁‘{𝑋})) | 
| 16 | ssel 3976 | . . . 4 ⊢ ((𝑁‘{𝑋}) ⊆ 𝑈 → (𝑋 ∈ (𝑁‘{𝑋}) → 𝑋 ∈ 𝑈)) | |
| 17 | 15, 16 | syl5com 31 | . . 3 ⊢ ((𝜑 ∧ 𝑋 ∈ 𝑉) → ((𝑁‘{𝑋}) ⊆ 𝑈 → 𝑋 ∈ 𝑈)) | 
| 18 | 17 | impr 454 | . 2 ⊢ ((𝜑 ∧ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈)) → 𝑋 ∈ 𝑈) | 
| 19 | 13, 18 | impbida 800 | 1 ⊢ (𝜑 → (𝑋 ∈ 𝑈 ↔ (𝑋 ∈ 𝑉 ∧ (𝑁‘{𝑋}) ⊆ 𝑈))) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ⊆ wss 3950 {csn 4625 ‘cfv 6560 Basecbs 17248 LModclmod 20859 LSubSpclss 20930 LSpanclspn 20970 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-riota 7389 df-ov 7435 df-0g 17487 df-mgm 18654 df-sgrp 18733 df-mnd 18749 df-grp 18955 df-lmod 20861 df-lss 20931 df-lsp 20971 | 
| This theorem is referenced by: ellspsn5b 20994 lsmelval2 21085 dihjat1lem 41431 | 
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